PARADISE LOST?
numbers are there? Are they more numerous than the integers,
or the rationals, or the algebraic numbers as a whole, or are they
less numerous? Since (by Cantor's theorem) the integers, the
rationals, and all algebraic numbers are equally numerous, the
question amounts to this: can the transcendental numbers be
counted off lf 2, 3, ...? Is the class of all transcendental
numbers similar to the class of all positive rational integers?
The answer is no; the transcendentals are infinitely more
numerous than the integers.
Here we begin to get into the controversial aspects of the
theory of sets. The conclusion just stated was like a challenge
to a man of Kronecker's temperament. Discussing Lindemann's
proof that TT is transcendental (see Chapter 24), Kronecker
asked, 'Of what use is your beautiful investigation regarding sr?
Why study such problems, since irrational [and hence trans-
cendental] numbers do not exist?* We can imagine the effect
on such a scepticism of Cantor's proof that the transcendentals
are infinitely more numerous than the integers 1, 2, 3, ...
which, according to Kronecker, are the noblest work of God
and the only numbers that do 'exist'.
Even a summary of Cantor's proof is out of the question here,
but something of the kind of reasoning he used can be seen
from the following simple considerations. If a class is similar
(hi the above technical sense) to the class of all positive
rational integers, the class is said to be denumerable^ The things*
in a denumerabie class can be counted off 1, 2, S, ...; the
things in a non-denumerable class can not be counted oft
1, 2, 8, ... : there will be more things in a non-denumerable
class than in a denumerable class. Do non-denumerable classes
exist? Cantor proved that they do* In fact the class of all points
on any line-segment, no matter how small the segment is
(provided it is more than a single point), is non-denumerable.
From this we see a hint of why the transcendentals are non-
denumerable. In the chapter on Gauss we saw that any root of
any algebraic equation is representable by a point on the plane
of Cartesian geometry. All these roots constitute the set of all
algebraic numbers, which Cantor proved to be denumerable.
But if the points on a mere line-segment are non-denumerable,
62T